Difference between revisions of "2016 AMC 12A Problems"
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==Problem 2== | ==Problem 2== | ||
− | + | For what value of <math>x</math> does <math>10^x \cdot 100^{2x} = 1000^5</math>? | |
− | For what value of <math>x</math> does <math>10^x\ | ||
<math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math> | <math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math> | ||
Line 18: | Line 17: | ||
==Problem 3== | ==Problem 3== | ||
+ | The remainder can be defined for all real numbers <math>x</math> and <math>y</math> with <math>y \neq 0</math> by <cmath>\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor</cmath>where <math>\left \lfloor \tfrac{x}{y} \right \rfloor</math> denotes the greatest integer less than or equal to <math>\tfrac{x}{y}</math>. What is the value of <math>\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )</math>? | ||
− | + | <math>\textbf{(A) } -\frac{3}{8} \qquad \textbf{(B) } -\frac{1}{40} \qquad \textbf{(C) } 0 \qquad \textbf{(D) } \frac{3}{8} \qquad \textbf{(E) } \frac{31}{40}</math> | |
− | |||
− | |||
− | |||
− | |||
− | |||
− | <math>\textbf{(A)} | ||
[[2016 AMC 12A Problems/Problem 3|Solution]] | [[2016 AMC 12A Problems/Problem 3|Solution]] | ||
Line 31: | Line 25: | ||
==Problem 4== | ==Problem 4== | ||
− | The mean, median, and mode of the 7 data values <math>60, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>? | + | The mean, median, and mode of the <math>7</math> data values <math>60, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>? |
<math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math> | <math>\textbf{(A)}\ 50\qquad\textbf{(B)}\ 60\qquad\textbf{(C)}\ 75\qquad\textbf{(D)}\ 90\qquad\textbf{(E)}\ 100</math> | ||
Line 51: | Line 45: | ||
==Problem 6== | ==Problem 6== | ||
− | A triangular array of 2016 coins in the first row, 2 coins in the second row, 3 coins in the third row, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math> ? | + | A triangular array of <math>2016</math> coins has <math>1</math> coin in the first row, <math>2</math> coins in the second row, <math>3</math> coins in the third row, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math> ? |
<math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math> | <math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 10</math> | ||
Line 64: | Line 58: | ||
\qquad\textbf{(B)}\ \text{two intersecting lines}\\ | \qquad\textbf{(B)}\ \text{two intersecting lines}\\ | ||
\qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\ | \qquad\textbf{(C)}\ \text{three lines that all pass through a common point}\\ | ||
− | \qquad\textbf{(D)}\ \text{three lines that do not all pass through a | + | \qquad\textbf{(D)}\ \text{three lines that do not all pass through a common point}\\ |
\qquad\textbf{(E)}\ \text{a line and a parabola}</math> | \qquad\textbf{(E)}\ \text{a line and a parabola}</math> | ||
+ | |||
[[2016 AMC 12A Problems/Problem 7|Solution]] | [[2016 AMC 12A Problems/Problem 7|Solution]] | ||
==Problem 8== | ==Problem 8== | ||
− | + | What is the area of the shaded region of the given <math>8\times 5</math> rectangle? | |
+ | |||
+ | <asy> | ||
+ | |||
+ | size(6cm); | ||
+ | defaultpen(fontsize(9pt)); | ||
+ | draw((0,0)--(8,0)--(8,5)--(0,5)--cycle); | ||
+ | filldraw((7,0)--(8,0)--(8,1)--(0,4)--(0,5)--(1,5)--cycle,gray(0.8)); | ||
+ | |||
+ | label("$1$",(1/2,5),dir(90)); | ||
+ | label("$7$",(9/2,5),dir(90)); | ||
+ | |||
+ | label("$1$",(8,1/2),dir(0)); | ||
+ | label("$4$",(8,3),dir(0)); | ||
+ | |||
+ | label("$1$",(15/2,0),dir(270)); | ||
+ | label("$7$",(7/2,0),dir(270)); | ||
− | <math>\textbf{(A)}\ | + | label("$1$",(0,9/2),dir(180)); |
+ | label("$4$",(0,2),dir(180)); | ||
+ | |||
+ | </asy> | ||
+ | |||
+ | <math>\textbf{(A)}\ 4.75\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 5.25\qquad\textbf{(D)}\ 6.5\qquad\textbf{(E)}\ 8</math> | ||
[[2016 AMC 12A Problems/Problem 8|Solution]] | [[2016 AMC 12A Problems/Problem 8|Solution]] | ||
+ | ==Problem 9== | ||
− | + | The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is <math>\tfrac{a-\sqrt{2}}{b}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math> ? | |
− | + | <asy> | |
+ | real x=.369; | ||
+ | draw((0,0)--(0,1)--(1,1)--(1,0)--cycle); | ||
+ | filldraw((0,0)--(0,x)--(x,x)--(x,0)--cycle, gray); | ||
+ | filldraw((0,1)--(0,1-x)--(x,1-x)--(x,1)--cycle, gray); | ||
+ | filldraw((1,1)--(1,1-x)--(1-x,1-x)--(1-x,1)--cycle, gray); | ||
+ | filldraw((1,0)--(1,x)--(1-x,x)--(1-x,0)--cycle, gray); | ||
+ | filldraw((.5,.5-x*sqrt(2)/2)--(.5+x*sqrt(2)/2,.5)--(.5,.5+x*sqrt(2)/2)--(.5-x*sqrt(2)/2,.5)--cycle, gray); | ||
+ | </asy> | ||
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 11</math> |
[[2016 AMC 12A Problems/Problem 9|Solution]] | [[2016 AMC 12A Problems/Problem 9|Solution]] | ||
− | |||
==Problem 10== | ==Problem 10== | ||
− | + | Five friends sat in a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</math> from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up? | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5</math> |
[[2016 AMC 12A Problems/Problem 10|Solution]] | [[2016 AMC 12A Problems/Problem 10|Solution]] | ||
− | |||
==Problem 11== | ==Problem 11== | ||
− | + | Each of the <math>100</math> students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are <math>42</math> students who cannot sing, <math>65</math> students who cannot dance, and <math>29</math> students who cannot act. How many students have two of these talents? | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 16\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 49\qquad\textbf{(E)}\ 64</math> |
[[2016 AMC 12A Problems/Problem 11|Solution]] | [[2016 AMC 12A Problems/Problem 11|Solution]] | ||
+ | ==Problem 12== | ||
− | == | + | In <math>\triangle ABC</math>, <math>AB = 6</math>, <math>BC = 7</math>, and <math>CA = 8</math>. Point <math>D</math> lies on <math>\overline{BC}</math>, and <math>\overline{AD}</math> bisects <math>\angle BAC</math>. Point <math>E</math> lies on <math>\overline{AC}</math>, and <math>\overline{BE}</math> bisects <math>\angle ABC</math>. The bisectors intersect at <math>F</math>. What is the ratio <math>AF</math> : <math>FD</math>? |
− | + | <asy> | |
+ | pair A = (0,0), B=(6,0), C=intersectionpoints(Circle(A,8),Circle(B,7))[0], F=incenter(A,B,C), D=extension(A,F,B,C),E=extension(B,F,A,C); | ||
+ | draw(A--B--C--A--D^^B--E); | ||
+ | label("$A$",A,SW); | ||
+ | label("$B$",B,SE); | ||
+ | label("$C$",C,N); | ||
+ | label("$D$",D,NE); | ||
+ | label("$E$",E,NW); | ||
+ | label("$F$",F,1.5*N); | ||
+ | </asy> | ||
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 3:2\qquad\textbf{(B)}\ 5:3\qquad\textbf{(C)}\ 2:1\qquad\textbf{(D)}\ 7:3\qquad\textbf{(E)}\ 5:2</math> |
[[2016 AMC 12A Problems/Problem 12|Solution]] | [[2016 AMC 12A Problems/Problem 12|Solution]] | ||
− | |||
==Problem 13== | ==Problem 13== | ||
− | + | Let <math>N</math> be a positive multiple of <math>5</math>. One red ball and <math>N</math> green balls are arranged in a line in random order. Let <math>P(N)</math> be the probability that at least <math>\tfrac{3}{5}</math> of the green balls are on the same side of the red ball. Observe that <math>P(5)=1</math> and that <math>P(N)</math> approaches <math>\tfrac{4}{5}</math> as <math>N</math> grows large. What is the sum of the digits of the least value of <math>N</math> such that <math>P(N) < \tfrac{321}{400}</math>? | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 12\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 20</math> |
[[2016 AMC 12A Problems/Problem 13|Solution]] | [[2016 AMC 12A Problems/Problem 13|Solution]] | ||
− | |||
==Problem 14== | ==Problem 14== | ||
− | + | Each vertex of a cube is to be labeled with an integer from <math>1</math> through <math>8</math>, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible? | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 24</math> |
[[2016 AMC 12A Problems/Problem 14|Solution]] | [[2016 AMC 12A Problems/Problem 14|Solution]] | ||
− | |||
==Problem 15== | ==Problem 15== | ||
− | + | Circles with centers <math>P, Q</math> and <math>R</math>, having radii <math>1, 2</math> and <math>3</math>, respectively, lie on the same side of line <math>l</math> and are tangent to <math>l</math> at <math>P', Q'</math> and <math>R'</math>, respectively, with <math>Q'</math> between <math>P'</math> and <math>R'</math>. The circle with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>? | |
− | <math>\textbf{(A)} | + | <math>\textbf{(A) } 0\qquad \textbf{(B) } \frac{\sqrt{6}}{3}\qquad\textbf{(C) } 1\qquad\textbf{(D) } \sqrt{6}-\sqrt{2}\qquad\textbf{(E) }\frac{\sqrt{6}}{2}</math> |
[[2016 AMC 12A Problems/Problem 15|Solution]] | [[2016 AMC 12A Problems/Problem 15|Solution]] | ||
− | |||
==Problem 16== | ==Problem 16== | ||
− | + | The graphs of <math>y=\log_3 x, y=\log_x 3, y=\log_\frac{1}{3} x,</math> and <math>y=\log_x \dfrac{1}{3}</math> are plotted on the same set of axes. How many points in the plane with positive <math>x</math>-coordinates lie on two or more of the graphs? | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6</math> |
[[2016 AMC 12A Problems/Problem 16|Solution]] | [[2016 AMC 12A Problems/Problem 16|Solution]] | ||
− | |||
==Problem 17== | ==Problem 17== | ||
− | + | Let <math>ABCD</math> be a square. Let <math>E, F, G</math> and <math>H</math> be the centers, respectively, of equilateral triangles with bases <math>\overline{AB}, \overline{BC}, \overline{CD},</math> and <math>\overline{DA},</math> each exterior to the square. What is the ratio of the area of square <math>EFGH</math> to the area of square <math>ABCD</math>? | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 1\qquad\textbf{(B)}\ \frac{2+\sqrt{3}}{3} \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{\sqrt{2}+\sqrt{3}}{2} \qquad\textbf{(E)}\ \sqrt{3}</math> |
[[2016 AMC 12A Problems/Problem 17|Solution]] | [[2016 AMC 12A Problems/Problem 17|Solution]] | ||
− | |||
==Problem 18== | ==Problem 18== | ||
− | + | For some positive integer <math>n,</math> the number <math>110n^3</math> has <math>110</math> positive integer divisors, including <math>1</math> and the number <math>110n^3.</math> How many positive integer divisors does the number <math>81n^4</math> have? | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 110\qquad\textbf{(B)}\ 191\qquad\textbf{(C)}\ 261\qquad\textbf{(D)}\ 325\qquad\textbf{(E)}\ 425</math> |
[[2016 AMC 12A Problems/Problem 18|Solution]] | [[2016 AMC 12A Problems/Problem 18|Solution]] | ||
− | |||
==Problem 19== | ==Problem 19== | ||
− | + | Jerry starts at <math>0</math> on the real number line. He tosses a fair coin <math>8</math> times. When he gets heads, he moves <math>1</math> unit in the positive direction; when he gets tails, he moves <math>1</math> unit in the negative direction. The probability that he reaches <math>4</math> at some time during this process is <math>\frac{a}{b},</math> where <math>a</math> and <math>b</math> are relatively prime positive integers. What is <math>a + b?</math> (For example, he succeeds if his sequence of tosses is <math>HTHHHHHH.</math>) | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 69\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\qquad\textbf{(E)}\ 313</math> |
[[2016 AMC 12A Problems/Problem 19|Solution]] | [[2016 AMC 12A Problems/Problem 19|Solution]] | ||
− | |||
==Problem 20== | ==Problem 20== | ||
− | + | A binary operation <math>\diamondsuit </math> has the properties that <math>a\ \diamondsuit\ (b\ \diamondsuit\ c) = (a\ \diamondsuit\ b)\cdot c</math> and that <math>a\ \diamondsuit\ a = 1</math> for all nonzero real numbers <math>a, b</math> and <math>c.</math> (Here the dot <math>\cdot</math> represents the usual multiplication operation.) The solution to the equation <math>2016\ \diamondsuit\ (6\ \diamondsuit\ x) = 100</math> can be written as <math>\frac{p}{q},</math> where <math>p</math> and <math>q</math> are relatively prime positive integers. What is <math>p + q?</math> | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 109\qquad\textbf{(B)}\ 201\qquad\textbf{(C)}\ 301\qquad\textbf{(D)}\ 3049\qquad\textbf{(E)}\ 33,601</math> |
[[2016 AMC 12A Problems/Problem 20|Solution]] | [[2016 AMC 12A Problems/Problem 20|Solution]] | ||
− | |||
==Problem 21== | ==Problem 21== | ||
− | + | A quadrilateral is inscribed in a circle of radius <math>200\sqrt{2}.</math> Three of the sides of this quadrilateral have length <math>200.</math> What is the length of its fourth side? | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 200\qquad\textbf{(B)}\ 200\sqrt{2} \qquad\textbf{(C)}\ 200\sqrt{3} \qquad\textbf{(D)}\ 300\sqrt{2} \qquad\textbf{(E)}\ 500</math> |
[[2016 AMC 12A Problems/Problem 21|Solution]] | [[2016 AMC 12A Problems/Problem 21|Solution]] | ||
− | |||
==Problem 22== | ==Problem 22== | ||
− | + | How many ordered triples <math>(x,y,z)</math> of positive integers satisfy <math>\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600</math> and <math>\text{lcm}(y,z)=900</math>? | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 15\qquad\textbf{(B)}\ 16\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 27\qquad\textbf{(E)}\ 64</math> |
[[2016 AMC 12A Problems/Problem 22|Solution]] | [[2016 AMC 12A Problems/Problem 22|Solution]] | ||
− | |||
==Problem 23== | ==Problem 23== | ||
− | + | Three numbers in the interval <math>\left[0,1\right]</math> are chosen independently and at random. What is the probability that the chosen numbers are the side lengths of a triangle with positive area? | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ \dfrac{1}{6}\qquad\textbf{(B)}\ \dfrac{1}{3}\qquad\textbf{(C)}\ \dfrac{1}{2}\qquad\textbf{(D)}\ \dfrac{2}{3}\qquad\textbf{(E)}\ \dfrac{5}{6}</math> |
[[2016 AMC 12A Problems/Problem 23|Solution]] | [[2016 AMC 12A Problems/Problem 23|Solution]] | ||
− | |||
==Problem 24== | ==Problem 24== | ||
− | + | There is a smallest positive real number <math>a</math> such that there exists a positive real number <math>b</math> such that all the roots of the polynomial <math>x^3-ax^2+bx-a</math> are real. In fact, for this value of <math>a</math> the value of <math>b</math> is unique. What is the value of <math>b?</math> | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12</math> |
[[2016 AMC 12A Problems/Problem 24|Solution]] | [[2016 AMC 12A Problems/Problem 24|Solution]] | ||
− | |||
==Problem 25== | ==Problem 25== | ||
− | + | Let <math>k</math> be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with <math>k+1</math> digits. Every time Bernardo writes a number, Silvia erases the last <math>k</math> digits of it. Bernardo then writes the next perfect square, Silvia erases the last <math>k</math> digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let <math>f(k)</math> be the smallest positive integer not written on the board. For example, if <math>k = 1</math>, then the numbers that Bernardo writes are <math>16, 25, 36, 49, 64</math>, and the numbers showing on the board after Silvia erases are <math>1, 2, 3, 4,</math> and <math>6</math>, and thus <math>f(1) = 5</math>. What is the sum of the digits of <math>f(2) + f(4)+ f(6) + ... + f(2016)</math>? | |
− | <math>\textbf{(A)}\ | + | <math>\textbf{(A)}\ 7986\qquad\textbf{(B)}\ 8002\qquad\textbf{(C)}\ 8030\qquad\textbf{(D)}\ 8048\qquad\textbf{(E)}\ 8064</math> |
[[2016 AMC 12A Problems/Problem 25|Solution]] | [[2016 AMC 12A Problems/Problem 25|Solution]] | ||
+ | |||
+ | ==See also== | ||
+ | |||
+ | {{AMC12 box|year=2016|ab=A|before=[[2015 AMC 12B Problems]]|after=[[2016 AMC 12B Problems]]}} | ||
+ | |||
+ | {{MAA Notice}} |
Latest revision as of 17:20, 19 January 2024
2016 AMC 12A (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See also
Problem 1
What is the value of ?
Problem 2
For what value of does
?
Problem 3
The remainder can be defined for all real numbers and
with
by
where
denotes the greatest integer less than or equal to
. What is the value of
?
Problem 4
The mean, median, and mode of the data values
are all equal to
. What is the value of
?
Problem 5
Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, ). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of?
Problem 6
A triangular array of coins has
coin in the first row,
coins in the second row,
coins in the third row, and so on up to
coins in the
th row. What is the sum of the digits of
?
Problem 7
Which of these describes the graph of ?
Problem 8
What is the area of the shaded region of the given rectangle?
Problem 9
The five small shaded squares inside this unit square are congruent and have disjoint interiors. The midpoint of each side of the middle square coincides with one of the vertices of the other four small squares as shown. The common side length is , where
and
are positive integers. What is
?
Problem 10
Five friends sat in a movie theater in a row containing seats, numbered
to
from left to right. (The directions "left" and "right" are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?
Problem 11
Each of the students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are
students who cannot sing,
students who cannot dance, and
students who cannot act. How many students have two of these talents?
Problem 12
In ,
,
, and
. Point
lies on
, and
bisects
. Point
lies on
, and
bisects
. The bisectors intersect at
. What is the ratio
:
?
Problem 13
Let be a positive multiple of
. One red ball and
green balls are arranged in a line in random order. Let
be the probability that at least
of the green balls are on the same side of the red ball. Observe that
and that
approaches
as
grows large. What is the sum of the digits of the least value of
such that
?
Problem 14
Each vertex of a cube is to be labeled with an integer from through
, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the cube are considered to be the same. How many different arrangements are possible?
Problem 15
Circles with centers and
, having radii
and
, respectively, lie on the same side of line
and are tangent to
at
and
, respectively, with
between
and
. The circle with center
is externally tangent to each of the other two circles. What is the area of triangle
?
Problem 16
The graphs of and
are plotted on the same set of axes. How many points in the plane with positive
-coordinates lie on two or more of the graphs?
Problem 17
Let be a square. Let
and
be the centers, respectively, of equilateral triangles with bases
and
each exterior to the square. What is the ratio of the area of square
to the area of square
?
Problem 18
For some positive integer the number
has
positive integer divisors, including
and the number
How many positive integer divisors does the number
have?
Problem 19
Jerry starts at on the real number line. He tosses a fair coin
times. When he gets heads, he moves
unit in the positive direction; when he gets tails, he moves
unit in the negative direction. The probability that he reaches
at some time during this process is
where
and
are relatively prime positive integers. What is
(For example, he succeeds if his sequence of tosses is
)
Problem 20
A binary operation has the properties that
and that
for all nonzero real numbers
and
(Here the dot
represents the usual multiplication operation.) The solution to the equation
can be written as
where
and
are relatively prime positive integers. What is
Problem 21
A quadrilateral is inscribed in a circle of radius Three of the sides of this quadrilateral have length
What is the length of its fourth side?
Problem 22
How many ordered triples of positive integers satisfy
and
?
Problem 23
Three numbers in the interval are chosen independently and at random. What is the probability that the chosen numbers are the side lengths of a triangle with positive area?
Problem 24
There is a smallest positive real number such that there exists a positive real number
such that all the roots of the polynomial
are real. In fact, for this value of
the value of
is unique. What is the value of
Problem 25
Let be a positive integer. Bernardo and Silvia take turns writing and erasing numbers on a blackboard as follows: Bernardo starts by writing the smallest perfect square with
digits. Every time Bernardo writes a number, Silvia erases the last
digits of it. Bernardo then writes the next perfect square, Silvia erases the last
digits of it, and this process continues until the last two numbers that remain on the board differ by at least 2. Let
be the smallest positive integer not written on the board. For example, if
, then the numbers that Bernardo writes are
, and the numbers showing on the board after Silvia erases are
and
, and thus
. What is the sum of the digits of
?
See also
2016 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by 2015 AMC 12B Problems |
Followed by 2016 AMC 12B Problems |
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All AMC 12 Problems and Solutions |
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